From 0975c25289d3d5066fc8a5faf693692411da8eb7 Mon Sep 17 00:00:00 2001 From: Claude Date: Thu, 23 Jul 2026 20:24:45 +0000 Subject: [PATCH 1/5] Add PLS model-inversion null-space score-plot figure New figure for the pid-book PLS model-inversion page: a two-component score plot of the cheddar-cheese model (trained on cheeses 5 to 30), inverted toward a taste of 20.9. Shows the calibration cheeses, the direct-inversion solution, the null space, and the O-PLS orthogonal space projected into the PLS score space; the null space and the projected orthogonal space coincide. Committed alongside its matplotlib generator, which imports process_improve. Co-Authored-By: Claude Opus 4.8 Claude-Session: https://claude.ai/code/session_01DsTypY4FcRbrXRxorb9raz --- pls/pls-model-inversion-null-space.png | Bin 0 -> 63230 bytes pls/pls-model-inversion-null-space.py | 123 +++++++++++++++++++++++++ 2 files changed, 123 insertions(+) create mode 100644 pls/pls-model-inversion-null-space.png create mode 100644 pls/pls-model-inversion-null-space.py diff --git a/pls/pls-model-inversion-null-space.png b/pls/pls-model-inversion-null-space.png new file mode 100644 index 0000000000000000000000000000000000000000..b2346a6faa2b67bb532ff992a5792cb51e60d87f GIT binary patch literal 63230 zcmdq}hdR8>v~-G`*}al<2;Vzyx%{6_7nr{PFfO)#Bf?gNs~mP z@FS7P+o&n=F9#FiJ@ALrb!FY_7aXjvyO}y!lFpc3ceHi5ZhO^ikE^AV^Hm3X@k64b zheU<;T)BST(OF7Z*zW&(#UTf$%fe0_6iav)8b=j9XA+6^9`TFpW2<};iL_vSTIqzA zd;EB}n*mGB3e|Me?hm)Mlr>Il@>JZcCG?`rb85n7{E;s6k&_|a>^9Bp=Hol{$C+3+ zu_})T-qKV$apKwHjiaSN;MGm-x577Rbo-cAV<@kn{Sr!snQS7k3A$M<18fyv93q?f8v9 z=Uw=pF-uv*_8Q)8p{ zm5th3m9D~T6b1I(3Q9_wQl37Ic>I`7O-=3X)$f$s_Nkv=QcKIs6dLb-zkQ@BvF5$o zR8m&f7Ee!4dJYNtr>Da1oY!5xwpUJ0ZkLQZ<5o@?c{4K}b#?VGnR?>ETymU~las0T ziTrogX(Ln>d71l15`C+O8y-g_Bpk{+e)up8KDOcx1IL&23;V}A^5__L9zK8KnLnfE zylwY8*0I*?Q?D-NP4|^kT)eAT7jgSGg{G!vj(xY3dc35@kA9W(@tFexJUry3YYQBt zKHrTMuHnWfbQdmMAo<!)?xWsFR_wt{JqKS#l0%~gNve4bq(+x*^O-34@oWF2E z@b#;s4n0Xa*+%Yb%SW}D zG$mzv^{?|TSuggVHDNU+-YbLEr|T%Y$yEgc>g(12jJI!-+gKI<_WirYdsn+m!}n}H zK0fYq!wk1>-@f+y8y~SG9UUFh^Yh9Z>uVQxOK?A9j+1f`Ff%io9{jZJ;$5B#mSNeF zzkmP6_9+_}#J^PJ9uks}V9U+Tt*xy^7>Qk6SYjFa`IDlW{rdImIp#H64qG?hd;h^m zO)aBsZEkj!M7nh861%jkP-FbD+kJf|o2aOMPL*z)e6a5fiR8I3UfxyY9AtZHcE>or z?w9Z1EA#C-8JU@HZ5KA`tKstXfAnalefPV7{>ryIa4&KH&NDec7Q; zpYFY}I2n?v7Hc-}@&3}%%u@jcK7TFEckkYP`SIhT3%l~m@fjusffl?qiS#2wM~A^OLdZ z!otGz?5skz(FXwmB_aogi$yx(SHr!C*^ahKGA0|#_J+&B{d^E09zvlanXGG2&AtAKKa$ZBZ*Zd`(EBSsP@M9$`yXL zLk843Iy%j-^taEC{gjhkilZg`El}qfT{MCJQZJQ#O`UwrZzmV5q`ke0h8I zLi(J5z3cZT-=k3zv9kk$cAb{je3U)@&hlkuX6BeyQj#_|jjMD-tk&MxB6#fB4h{)h zzxVH@$0sICZES+J?3U_FOY~7tR}Zc?Ka(J{1DX4wtoOV62?|R3zR=%OQ^I-2d3pUi zId|@?>(q{z3JMn8CGARo+;hSI$&=kXkJ<+#B}cbUlok~g72cQ^uxd&O!OA6{xJ~)O z@V!KQe7uNV$KkH`ZrMr`jgOAd;r$Kd^j}-tZ?^HqDHjV;6k_}N@%}b;sq6eZk68O( z%Ck=Pm>d4}BV)UEmcfDfodO(*9})dIUVojgkGC-q>+gdd%|1F0n{l`LYKk!E~(U6^VA)>RRc!*VLPCnT@e< z&Zw!VgmD>v?E zQqVlh&MgoX6I0#%mDSDb`HWhEpa5P0Ou7eBJ_+28u7`}J4L~Z=JKJoMCg+q1*r9OuCtE7im zT{;??0}E1j{QYzAb*C5Xy5Hp~WZzYCcbBwjf1TRp-jQ#oZ$bYrdNz7}HyREfIWjaU z?J+y(bpJh0TK%_gkuKwH8jTq|#(ICuFB+5{)Xx_o)pvAUvQn=J;}U*tsN9(Gc1_dB z%kt}pnkS-WcJOrW)B&WlkqKv1TAcqpp0TlCzj8`9*7ZkqI}02xlO9G#M>ppO2h;cb zJ!o8F@Ujtg!(!4Hp`oXzmx+)U3rkK;E?-|?amf0BcQ`Ojt1>q|&A;GPvN9>1+PVAq zjqAA+Jtf}n)dy

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zoc~kju=>lxoAZ#{0ko+Y*b>E@ydv`)`n_}3@|koEP43icZ6aX}JT1bp51E`5!COYX z&>|XoqM5^B#&uCr1^D+D+onSGtNzCiIq)slE9h;(n|_6hGV0Awr=Y1k`3P)jP!}*} zbb$SGtw=IK#Ck2;{%YRW+GBY{?T)Wcs}?H{b3KQ-1+%IcU89eQu8Qy5w-3#e1#PEI ztb?z=15&}CI{$EAS<}PXC3abxo06g#)!)BAA8U<%`}2UV$V#6TJ&(TE=H{k#6*d1D zEE1v=Ity~k!2x^#W7UcHJz zmop~U&1hf;ERhLPrH(6~0j|Oep9a$fG z8{+KozP%{$fjI32Hsb$hKlyvnJ&;2q0rtRHDd@_58hRC2oYsO}yo~ws?u%P5q_ASK|YK^3HheIFhvnFk^%6dkkt<(4ME1hQSgfcU#aJ?@-^!M z*r5n}u6#TN+z0AhZQ%}^2)m%*dRQ$#EaY%zeAY$8jqBO&7vHj({33`W#|^sfXrcQ% zm`l-|IB^jJpo&pnVb(VqO2ml*>ImROVq%h(Z$6me<>j?twwBv%0w3WHEz)E{>8t6W z#tc0x=Av1?vt10?rgP`4?eFi0N;js9(P;~l2(5zeNr9djXbloE%eI?)hEZ6V?r(9} zoGKXXght0p;4+jX8OSz$jT)jv)NUBzR|E0|GwRY|AhO*q#Z{Dp4$0pu$3PZqJq@vp z_L2va7&DS4xy&+#Od|AR)uAfioib;`i&zGGR%pt9D?-4NhNjkdfZR--KR> zPj*KEBPr?EZ$^A~{8GTjk5S+|bSv{P*mg6bF;;~%w4sla;wn``y-Bbe5uE6|jH(51*3I>2j zV?AMhwF4#*uVGd+;x+f None: + """Fit the models, invert, and render the score plot.""" + cheese = pd.read_csv(DATA_URL) + train = cheese.iloc[4:] # cheeses 5 to 30 + x_train = train[X_COLUMNS] + y_train = train[["Taste"]] + + pls = PLS(n_components=2).fit(x_train, y_train) + opls = OPLS(n_orthogonal_components=1).fit(x_train, y_train) + + result = pls.invert(y_desired=TARGET_TASTE) + tau = result.scores.to_numpy() + g_ns = result.null_space_basis.to_numpy().ravel() + + # Null-space line in the PLS score space. + steps = np.linspace(-4.0, 4.0, 60) + ns_line = np.array([tau + s * g_ns for s in steps]) + + # O-PLS orthogonal space: walk along the orthogonal loading in the input + # space (scaled), starting from the O-PLS design, then project into the PLS + # score space with the direct weights. These points should land on the NS. + opls_result = opls.invert(y_desired=TARGET_TASTE) + x_scaler = pls._x_scaler # noqa: SLF001 - reuse the fitted scaling for projection + x_design_scaled = x_scaler.transform(opls_result.x_new.to_frame().T).to_numpy().ravel() + os_direction = opls_result.orthogonal_space_basis.to_numpy().ravel() + os_direction = os_direction / np.linalg.norm(os_direction) + direct_weights = pls.direct_weights_.to_numpy() # W* maps scaled X -> scores + os_points = np.array( + [(x_design_scaled + s * os_direction) @ direct_weights for s in np.linspace(-3.0, 3.0, 25)] + ) + + scores = pls.scores_.to_numpy() + + fig, ax = plt.subplots(figsize=(6.4, 5.2)) + ax.scatter(scores[:, 0], scores[:, 1], color=DARK_BLUE, s=30, label="Calibration cheeses") + ax.plot(ns_line[:, 0], ns_line[:, 1], color=PURPLE, lw=2, label="Null space (PLS inversion)") + ax.scatter( + os_points[:, 0], + os_points[:, 1], + facecolors="none", + edgecolors=GREEN, + s=42, + label="Orthogonal space (O-PLS), projected", + ) + ax.scatter( + tau[0], tau[1], color=BLACK, marker="s", s=70, zorder=5, label="Direct-inversion solution" + ) + ax.axhline(0, color="0.7", lw=0.8) + ax.axvline(0, color="0.7", lw=0.8) + ax.set_xlabel("$t_1$") + ax.set_ylabel("$t_2$") + ax.set_title(f"Cheddar cheese: designing toward a taste of {TARGET_TASTE}") + ax.legend(loc="best", framealpha=0.9, fontsize=9) + fig.tight_layout() + fig.savefig(out_dir / "pls-model-inversion-null-space.png") + plt.close(fig) + + +def main() -> None: + """Entry point: render into the given directory (default: this script's own).""" + out_dir = Path(sys.argv[1]) if len(sys.argv) > 1 else Path(__file__).resolve().parent + build_figure(out_dir) + + +if __name__ == "__main__": + main() From a85834365af32285d4899598a9cd1f4e8690bd95 Mon Sep 17 00:00:00 2001 From: Claude Date: Thu, 23 Jul 2026 21:23:37 +0000 Subject: [PATCH 2/5] Restyle inversion figure: orange null space, bolder O-PLS circles, second target Recolours the main null-space line to orange, enlarges the O-PLS orthogonal-space circles and thickens their edge, and adds a red dashed null-space line for a second inversion (held-out cheese 4, taste 47.9). The two null-space lines are parallel, since a single-response model has one null-space direction and only the offset changes with the target. 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taste. In a single-response + # model every target shares the same null-space direction, so this line is + # parallel to the first, just shifted to pass through its own solution. + result2 = pls.invert(y_desired=SECOND_TASTE) + tau2 = result2.scores.to_numpy() + ns_line2 = np.array([tau2 + s * g_ns for s in steps]) + # O-PLS orthogonal space: walk along the orthogonal loading in the input # space (scaled), starting from the O-PLS design, then project into the PLS # score space with the direct weights. These points should land on the NS. @@ -90,13 +99,22 @@ def build_figure(out_dir: Path) -> None: fig, ax = plt.subplots(figsize=(6.4, 5.2)) ax.scatter(scores[:, 0], scores[:, 1], color=DARK_BLUE, s=30, label="Calibration cheeses") - ax.plot(ns_line[:, 0], ns_line[:, 1], color=PURPLE, lw=2, label="Null space (PLS inversion)") + ax.plot(ns_line[:, 0], ns_line[:, 1], color=ORANGE, lw=2, label=f"Null space (taste {TARGET_TASTE})") + ax.plot( + ns_line2[:, 0], + ns_line2[:, 1], + color=RED, + lw=2, + linestyle="--", + label=f"Null space (taste {SECOND_TASTE})", + ) ax.scatter( os_points[:, 0], os_points[:, 1], facecolors="none", edgecolors=GREEN, - s=42, + s=70, + linewidths=1.8, label="Orthogonal space (O-PLS), projected", ) ax.scatter( From f85ac6cfebefe073c6e2837f627ad08e1da26c3e Mon Sep 17 00:00:00 2001 From: Claude Date: Thu, 23 Jul 2026 21:31:08 +0000 Subject: [PATCH 3/5] Add the lowest-taste inversion (12.3) as a third, parallel null-space line Adds a purple dotted null-space line for the lowest held-out taste (cheese 1, 12.3), in a cool colour contrasting the warm orange and red. All three lines are exactly parallel (shared direction g = [0.433, 0.902]); only the offset shifts with the target, confirming that a single-response model has one fixed null-space direction. 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into the PLS # score space with the direct weights. These points should land on the NS. @@ -108,6 +114,14 @@ def build_figure(out_dir: Path) -> None: linestyle="--", label=f"Null space (taste {SECOND_TASTE})", ) + ax.plot( + ns_line3[:, 0], + ns_line3[:, 1], + color=PURPLE, + lw=2, + linestyle=":", + label=f"Null space (taste {THIRD_TASTE})", + ) ax.scatter( os_points[:, 0], os_points[:, 1], From 9515a26e246d835744778b514189696865e60575 Mon Sep 17 00:00:00 2001 From: Claude Date: Thu, 23 Jul 2026 21:31:59 +0000 Subject: [PATCH 4/5] Reorder figure legend: null-space lines first, then markers, top right Co-Authored-By: Claude Opus 4.8 Claude-Session: https://claude.ai/code/session_01DsTypY4FcRbrXRxorb9raz --- pls/pls-model-inversion-null-space.png | Bin 83859 -> 83360 bytes pls/pls-model-inversion-null-space.py | 12 +++++++++++- 2 files changed, 11 insertions(+), 1 deletion(-) diff --git a/pls/pls-model-inversion-null-space.png b/pls/pls-model-inversion-null-space.png index 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zQ;QFcUEe&ACqgfq9}aemmhmF65Qa8WslH7gy@31+jEfGi)PWosmR43+e_eDj{QZJD zFH2=*WhIh2pBF#ZYkE@LMH<(P!NRF4wM6}e>({R%F=yhlws#+us=;m;9(Rp7YtV!; zB>T0Ci>JZ~KVm_#Zwj#y4$W;@MgdR=o52 QCWWt%vEf1Q2E;A>2f`t)(*OVf diff --git a/pls/pls-model-inversion-null-space.py b/pls/pls-model-inversion-null-space.py index f3f7c8b..7e42383 100644 --- a/pls/pls-model-inversion-null-space.py +++ b/pls/pls-model-inversion-null-space.py @@ -139,7 +139,17 @@ def build_figure(out_dir: Path) -> None: ax.set_xlabel("$t_1$") ax.set_ylabel("$t_2$") ax.set_title(f"Cheddar cheese: designing toward a taste of {TARGET_TASTE}") - ax.legend(loc="best", framealpha=0.9, fontsize=9) + # Reorder the legend so the lines (null spaces) come first, then the markers + # (calibration cheeses, orthogonal space, direct-inversion solution). + handles, labels = ax.get_legend_handles_labels() + order = [1, 2, 3, 0, 4, 5] + ax.legend( + [handles[i] for i in order], + [labels[i] for i in order], + loc="upper right", + framealpha=0.9, + fontsize=9, + ) fig.tight_layout() fig.savefig(out_dir / "pls-model-inversion-null-space.png") plt.close(fig) From d890cc39a5bffb65493fe6da3c6dc8c94a122e67 Mon Sep 17 00:00:00 2001 From: Claude Date: Thu, 23 Jul 2026 21:45:21 +0000 Subject: [PATCH 5/5] Split into two legends: null-space lines top left, markers bottom right The lines legend lists the null spaces from lowest to highest taste (12.3, 20.9, 47.9); the markers legend holds the calibration cheeses, the O-PLS orthogonal space, and the direct-inversion solution. 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z`!f}*0pjtEY-&i!(YUjQ>yAQD~Wec4cnp2|*5QVEJHp;=cKVP1cETzR`0e=*a#s6NX#PzFEbV|m9Z4Og?f5+CiTt5DQw zEAr5NvVvs#VvgV+SW2LOl8S3d6Eg(o@qqr76&PJQpOHZ|&;!xKP8slPV~1Eb8jU9E zkt5FSAQ6?025jQPWo{(WJX33HAAo!ZuqG!rR}l6z5A@1A8(TdHOV#O+Nr|;{Uqp)c z?7EjywVq1ddHk_RF~G0u-{rMHGO9)L{#x!wA}zRxak>bU>~P8Fw1yB*&$B>!5uy;{ zGm)6|o6nd>dP1FR7gQd8yoAQkOsIEC(}28Pza1+UZvVOH?`7l{mY`9j6jF6Qh~FlJ?o@nl%Fr*CjrvQL#J{K|Yr!Og pdjH|^es2Fygq`Sv{1@$lnHBDmQETz$XENgFG>;uy%Qqi5^)Kv&AiMwo diff --git a/pls/pls-model-inversion-null-space.py b/pls/pls-model-inversion-null-space.py index 7e42383..1cb483c 100644 --- a/pls/pls-model-inversion-null-space.py +++ b/pls/pls-model-inversion-null-space.py @@ -104,25 +104,17 @@ def build_figure(out_dir: Path) -> None: scores = pls.scores_.to_numpy() fig, ax = plt.subplots(figsize=(6.4, 5.2)) - ax.scatter(scores[:, 0], scores[:, 1], color=DARK_BLUE, s=30, label="Calibration cheeses") - ax.plot(ns_line[:, 0], ns_line[:, 1], color=ORANGE, lw=2, label=f"Null space (taste {TARGET_TASTE})") - ax.plot( - ns_line2[:, 0], - ns_line2[:, 1], - color=RED, - lw=2, - linestyle="--", - label=f"Null space (taste {SECOND_TASTE})", + cal = ax.scatter(scores[:, 0], scores[:, 1], color=DARK_BLUE, s=30, label="Calibration cheeses") + (line3,) = ax.plot( + ns_line3[:, 0], ns_line3[:, 1], color=PURPLE, lw=2, linestyle=":", label=f"Null space (taste {THIRD_TASTE})" ) - ax.plot( - ns_line3[:, 0], - ns_line3[:, 1], - color=PURPLE, - lw=2, - linestyle=":", - label=f"Null space (taste {THIRD_TASTE})", + (line1,) = ax.plot( + ns_line[:, 0], ns_line[:, 1], color=ORANGE, lw=2, label=f"Null space (taste {TARGET_TASTE})" ) - ax.scatter( + (line2,) = ax.plot( + ns_line2[:, 0], ns_line2[:, 1], color=RED, lw=2, linestyle="--", label=f"Null space (taste {SECOND_TASTE})" + ) + os_circles = ax.scatter( os_points[:, 0], os_points[:, 1], facecolors="none", @@ -131,7 +123,7 @@ def build_figure(out_dir: Path) -> None: linewidths=1.8, label="Orthogonal space (O-PLS), projected", ) - ax.scatter( + di = ax.scatter( tau[0], tau[1], color=BLACK, marker="s", s=70, zorder=5, label="Direct-inversion solution" ) ax.axhline(0, color="0.7", lw=0.8) @@ -139,17 +131,14 @@ def build_figure(out_dir: Path) -> None: ax.set_xlabel("$t_1$") ax.set_ylabel("$t_2$") ax.set_title(f"Cheddar cheese: designing toward a taste of {TARGET_TASTE}") - # Reorder the legend so the lines (null spaces) come first, then the markers - # (calibration cheeses, orthogonal space, direct-inversion solution). - handles, labels = ax.get_legend_handles_labels() - order = [1, 2, 3, 0, 4, 5] - ax.legend( - [handles[i] for i in order], - [labels[i] for i in order], - loc="upper right", - framealpha=0.9, - fontsize=9, + + # Two legends: the null-space lines (lowest to highest taste) at top left, and + # the markers at bottom right. + lines_legend = ax.legend( + handles=[line3, line1, line2], loc="upper left", framealpha=0.9, fontsize=9, title="Null spaces" ) + ax.add_artist(lines_legend) + ax.legend(handles=[cal, os_circles, di], loc="lower right", framealpha=0.9, fontsize=9) fig.tight_layout() fig.savefig(out_dir / "pls-model-inversion-null-space.png") plt.close(fig)